The local consumer watchdog group was concerned about the cost of electricity to residential customers over the New Zealand winter months (Southern Hemisphere). They took a random sample of 25 residential electricity accounts and looked at the total cost of electricity used over the three months of June, July, and August. The costs were:

Assume that the amount of electricity used over the three months by a residential account is normal \(\left({ }^{2}\right.\) ), where the known standard deviation \(=80\).
(a) Use a normal \(\left(32580^{2}\right)\) prior for . Find the posterior distribution for .
(b) Find a 95\% Bayesian credible interval for .
(c) Perform a Bayesian test of the hypothesis
\[
H_{0}:=350 \text { versus } H_{1}:=350
\]
at the \(5 \%\) level.

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